Forthcoming mobile communication systems are foreseen to provide ubiquitous connectivity and seamless service delivery in all circumstances. The large number of devices and the coexistence of human-centric and machine type applications expected will lead to a large diversity of communication scenarios and characteristics. In this context, many advanced communication techniques are under investigation. Each of these techniques is typically suitable for a subset of the foreseen communication scenarios.
One category of techniques is based on filter-bank multicarrier communications principles. A filter-bank multicarrier (FBMC) communications system is composed of a synthesis filter for the modulation and an analysis filter for demodulation. The synthesis and analysis filters are composed of M channels, denoted by sub-carriers for a communications system. The channel number m of the synthesis filter modulates a complex signal cn(m) carrying information at time slot n. This channel consists of an oversampling operation by N followed by a finite impulse response filter Fm(z). These operations can be expressed as follows:
                              Upsampling          ⁢                                          ⁢          by          ⁢                                          ⁢          N                ⁢                                  ⁢                                            C                                                ↑                  N                                ,                k                                      ⁡                          (              m              )                                =                      {                                                                                                                                                        C                                                      k                            N                                                                          ⁡                                                  (                          m                          )                                                                    ⁢                      if                      ⁢                                                                                          ⁢                                                                        mod                          N                                                ⁡                                                  (                          kN                          )                                                                                      =                    0                                                                                                0                                                                                        (        1        )                                          Filter          ⁢                                          ⁢          by          ⁢                                          ⁢                                    F              m                        ⁡                          (              z              )                                      ⁢                                  ⁢                                            s              m                        ⁡                          (              k              )                                =                                    ∑                              l                =                                  -                  ∞                                                            +                ∞                                      ⁢                                                  ⁢                                                            C                                                            ↑                      N                                        ,                                          k                      -                      l                                                                      ⁡                                  (                  m                  )                                            ⁢                                                f                  m                                ⁡                                  (                  l                  )                                                                                        (        2        )                                                      f            m                    ⁡                      (            l            )                          =                              g            ⁡                          (              l              )                                ⁢                      e                                          i                ⁢                                                                  ⁢                2                ⁢                                                                  ⁢                                  π                  ⁢                  lm                                            M                                                          (        3        )            
g is called the prototype filter, and is a function with a finite lengthL:g(k)=0 if k∉0,L−1.  (4)
The modulated signal s(k) at the output of the synthesis filter is obtained after the sum of each channel output:
                              s          ⁡                      (            k            )                          =                              ∑                          m              =              0                                      M              -              1                                ⁢                                          ⁢                                    s              m                        ⁡                          (              k              )                                                          (        5        )            
An equivalent relation between the input and output signal of the synthesis filter can be expressed as follows:
                              s          ⁡                      (            k            )                          =                              ∑                          m              =              0                                      M              -              1                                ⁢                                          ⁢                                    ∑                              n                =                                  -                  ∞                                                            +                ∞                                      ⁢                                                  ⁢                                                            c                  n                                ⁡                                  (                  m                  )                                            ⁢                              g                ⁡                                  (                                      k                    -                    nN                                    )                                            ⁢                              e                                                      i                    ⁢                                                                                  ⁢                    2                    ⁢                                          π                      ⁢                      km                                                        M                                                                                        (        6        )            
This expression leads to the polyphase network representation of the synthesis filter, which is computationally complex to implement when compared to a direct synthesis filter representation.
Concerning the analysis filter, the dual operations of the transmitter for the M channels are comprised of the following operation:
                                          Filtering            ⁢                                                  ⁢            by            ⁢                                                  ⁢                                          H                m                            ⁡                              (                z                )                                      ⁢                          :                        ⁢                                                  ⁢                                          r                m                            ⁡                              (                k                )                                              =                                    ∑                              l                =                                  -                  ∞                                                            +                ∞                                      ⁢                                                  ⁢                                          r                ⁡                                  (                                      k                    -                    1                                    )                                            ⁢                                                h                  m                                ⁡                                  (                  l                  )                                                                    ,                                            h              m                        ⁡                          (              l              )                                =                                    g              ⁡                              (                l                )                                      ⁢                          e                                                                    -                    i                                    ⁢                                                                          ⁢                  2                  ⁢                  π                  ⁢                                                                          ⁢                  lm                                M                                                                        (        7        )                                          Downsampling          ⁢                                          ⁢          by          ⁢                                          ⁢          N          ⁢                      :                    ⁢                                          ⁢                                    d              n                        ⁡                          (              m              )                                      =                              r            m                    ⁡                      (            nN            )                                              (        8        )            
The equivalent relation between the input and output signals of the analysis filter can be expressed as follows:
                                          d            n                    ⁡                      (            m            )                          =                              ∑                          k              =                              -                ∞                                                    +              ∞                                ⁢                                          ⁢                                    s              ⁡                              (                k                )                                      ⁢                          g              ⁡                              (                                  k                  -                  nN                                )                                      ⁢                          e                                                                    -                    i                                    ⁢                                                                          ⁢                  2                  ⁢                                                                          ⁢                                      π                    ⁢                    km                                                  M                                                                        (        9        )            
This basic approach is used in numerous modulation schemes, and as such the design of filters implementing the described approach is an important activity.
This definition of the FBMC technique corresponds to numerous modulation schemes depending on the design of the underlined prototype filters and the choice of the different set of parameters (M, N, L, . . . ). One such modulation scheme is Filter-Bank Multi-Carrier with Offset Quadrature Amplitude Modulation (FBMC/OQAM) which is being studied and considered as a key enabler for the future flexible 5G air interface for example. It exhibits better spectrum shape compared to the traditional Orthogonal Frequency-Division Multiplexing (OFDM) and enables better spectrum usage and improved mobility support. This is possible thanks to the use of a prototype filter which makes it possible to improve the time and frequency localization properties of the transceiver. Orthogonality is preserved in the real field with the OQAM scheme. FBMC/OQAM implementation is not different from OFDM as it relies on Fast Fourier Transform (FFT) processing with an additional low-complexity PolyPhase Network (PPN) filtering stage. However, the choice of the prototype filter is crucial for FBMC/OQAM modulation, as the time/frequency localization of this filter can significantly impact the different performance levels and the frame structure of the communication system. Furthermore, the length of the prototype filter impacts considerably the receiver complexity. Thus, design of new filters is of high interest to improve robustness of FBMC/OQAM against channel impairments and to support the constraints imposed by various 5G scenarios while preserving reasonable receiver complexity.
FBMC is a multicarrier transmission scheme that introduces a filter-bank to enable efficient pulse shaping for the signal conveyed on each individual subcarrier. This additional element represents an array of band-pass filters that separate the input signal into multiple components or subcarriers, each one carrying a single frequency sub-band of the original signal. The process of decomposition performed by the filter bank is called analysis (meaning analysis of the signal in terms of its components in each sub-band); the output of analysis is referred to as a sub-band signal with as many sub-bands as there are filters in the filter bank. The reconstruction process is called synthesis, meaning reconstitution of a complete signal resulting from the filtering process. Such a transceiver structure usually requires a higher implementation complexity related not only to the filtering steps but also to the applied modifications to the modulator/demodulator architecture. However, the usage of digital polyphase filter bank structures together with the rapid growth of digital processing capabilities in recent years have made FBMC a practically feasible approach. As a promising variant of filtered modulation schemes, FBMC/OQAM, (also called OFDM/OQAM or staggered modulated multitone—SMT), can usually achieve a higher spectral efficiency than OFDM since it does not require the insertion of a Cyclic-Prefix (CP). Additional advantages include the robustness against highly variant fading channel conditions and imperfect synchronizations by selecting the appropriate prototype filter type and coefficients. 4G/LTE is based on OFDM multicarrier modulation. In accordance with the Balian-Low theorem, OFDM:
1) respects the complex orthogonality,
2) is poorly localized in frequency domain by adopting a rectangular waveform,
3) wastes part of the available bandwidth due to the addition of a CP.
Property 2 results in high Out-Of-Band Power Leakage (OOBPL), and large guard-bands have to be inserted to respect Adjacent Channel Leakage Power Ratio (ACLR) requirements. Furthermore, it results in a poor robustness against Doppler shift and spread. Further possible disadvantages of the corresponding OFDM system are related to flexible spectrum usage scenarios, where spectrum sharing and fragmented usage are not efficiently supported.
To overcome the shortcomings 2) and 3) of OFDM, FBMC/OQAM:
a) relaxes to real field orthogonality,
b) is better localized in time and frequency, depending on the used prototype filter,
c) uses efficiently available bandwidth to achieve a higher spectral efficiency.
Property a) is obtained by changing the way QAM symbols are mapped onto each subcarrier. Instead of sending a complex symbol (I and Q) of duration T as in classical CP-OFDM, the real and imaginary parts are separated and sent with an offset of T/2 (hence the name Offset-QAM). Improvement b) comes from the introduction of the filter-bank and therefore highly depends on its type and coefficients. Property c) is the consequence of the absence of a CP. Previous published works have identified two major design criteria for an FBMC/OQAM system:                Time Frequency Localization (TFL) criterion: for a better localized waveform in time and frequency domains thanks to the prototype filter. It is predictable that FBMC systems exhibit better robustness than CP-OFDM in doubly-dispersive channels and in the case of communications with synchronization errors, as described in “A survey on multicarrier communications: Prototype filters, lattice structures and implementation aspects”, A. Sahin et al., IEEE communications surveys & Tutorials, vol. 16, No 3, pp 1312-1338, 2014. To this purpose filter designs with the optimized TFL criterion have been proposed, such as Isotropic Orthogonal Transform Algorithm with overlapping factor (OF) equals to 4.        Lower sideband criterion: for achieving low out-of-band power leakage in frequency domain and for improving spectrum coexistence with other systems. To this purpose, particular filter types should be used such as Martin-Mirabassi-Bellange with OF equals to 4, as considered for FBMC/OQAM during the PHYDYAS project, as described at http://www.ict-phydyas.org/.FBMC/OQAM System Description        
Two basic implementation approaches for the FBMC/OQAM modulation exist, each having different characteristics in terms of computational or hardware complexity and performance.
FIG. 1 shows a PPN FBMC/OQAM transmitter implementation.
As shown, the implementation of FIG. 1 comprises an OQAM mapper 110 comprising a QAM mapper 111 creating real and imaginary values from a binary input. The imaginary values are delayed by T/2 with respect to the real values by delay unit 112. The real and imaginary values are output to respective processing channels. Each processing channel comprises in sequence a preprocessing unit 121, 122, an Inverse Fast Fourier Transform (IFFT) unit 131, 132 and a Polyphase Network (PPN) 141, 142. The outputs of the two processing channels are then combined by a summer 150.
In operation, a Pulse-amplitude modulation (PAM) symbol at subcarrier index m and time slot n, an(m) is obtained from the QAM mapper 111 equivalent to OFDM, where M is the total number of available sub-carriers and Ns the number of FBMC symbols with separation of real and imaginary parts respectively at time slots 2n and 2n+1.
Pre-processing unit 121, 122 computes the phase term ϕn(m), which to keep the orthogonality in the real field must be a quadrature phase rotation term, i.e.ϕn(m)=jn+m, or ϕn(m)=jn+m−n m.
The outputs of two Inverse Fast Fourier Transform (IFFT) blocks 131, 132 separately process the real and imaginary sub-carriers in accordance with the equation:
                                          u            n                    ⁡                      (            k            )                          =                              ∑                          m              =              0                                      M              -              1                                ⁢                                          ⁢                                                    a                n                            ⁡                              (                m                )                                      ⁢                                          ϕ                n                            ⁡                              (                m                )                                      ⁢                          e                              j                ⁢                                                      2                    ⁢                                                                                  ⁢                    π                                    M                                ⁢                km                                                                        (        10        )            
The Polyphase Network (PPN) 141, 142 corresponds in effect to a bank of phase shifters providing a respective phase shift for each sub-carrier. Specifically, the Polyphase Network filter decomposition is described as follows:
                                          s            l                    ⁡                      (            k            )                          =                              ∑                          n              =              0                                                      N                a                            -              1                                ⁢                                          ⁢                                    g              ⁡                              (                                  k                  -                  nM                                )                                      ⁢                                          u                                                      2                    ⁢                    n                                    +                  1                                            ⁡                              (                k                )                                                                        (        11        )            
The PPN 141, 142 can be seen as a digital filter, whose design represents important trade-offs between performance and system complexity and the final overlap and sum due to OQAM processing.
                                          s            l                    ⁡                      (            k            )                          =                                            s              0                        ⁡                          (              k              )                                +                                    s              1                        ⁡                          (                              k                -                                  M                  2                                            )                                                          (        12        )            
When using a short filter with an Overlapping Factor of 1 for the PPN, this latter can be seen as a windowing operation: the outputs of the IFFT are simply multiplied by the prototype filter impulse response. Consequently, the hardware complexity overhead introduced by the PPN is limited.
Secondly, a Frequency Spread (FS) implementation is possible.
FIG. 2a shows an FS FBMC/OQAM transmitter implementation. As shown, the implementation of FIG. 2 comprises an OQAM mapper 210 comprising a QAM mapper 211 creating real and imaginary values from a binary input. The imaginary values are delayed by T/2 with respect to the real values by delay unit 212. The real and imaginary values are output to respective processing channels. Each processing channel comprises in sequence a preprocessing unit 221, 222, an upsampling unit 231, 232, which upscale each signal by a factor of q, a Finite Impulse Response (FIR) filter 241, 242 and an Inverse Fast Fourier Transform block 251, 252. The outputs of the two processing channels are then combined by a summer 260.
The original concept of this design is to shift the filtering stage into the frequency domain, to enable the use of a low-complexity per-sub-carrier equalizer as in OFDM. The hardware complexity is supposed to be higher than the complexity of the PPN implementation, at least for long filters. In fact, it requires one FFT of size L=qM per OQAM symbol, where q is the overlapping factor, and M the total number of available sub-carriers. However, in the case of a short filter (q=1), the size of the FFT is the same as for the PPN implementation.
FIG. 2b shows an FS FBMC/OQAM Receiver implementation. As shown, a received signal is sampled by a first sliding window 263. The received signal is furthermore subjected to a M/2 delay by delay unit 270, where M is the length of the first sliding window 263 and also a second sliding window 264 to which the output of the delay unit 270 is fed, so that the sampled periods of the two sliding windows 263, 264 overlap by half their respective lengths. Each Sliding window 263, 264 outputs samples to respective Fast Fourier Transform units 253, 254. Fast Fourier Transform units 253, 254 provide their outputs to respective Digital Filters 243, 244, the outputs of which are then down sampled by downsampling units 233, 234 which down sample by a factor of q. The outputs of the downsampling units 233, 234 are processed in respective post processing units 223, 224, and then finally the real components are mapped back to recover the original data by the QAM demapper 281.
Typical FBMC/OQAM architectures use a prototype filter with a duration 4 times higher than an OFDM symbol. However, a shorter filter can also be applied, such as the Quadrature Mirror Filter with OF equal to 1 which was recently applied to FBMC/OQAM leading to a variant denoted by Lapped-OFDM modulation.
It is desirable to find short prototype filter designs with good performance and low hardware complexity, and methodologies for developing such designs.